90,002 research outputs found
Orthogonal weighted linear L1 and L∞ approximation and applications
AbstractLet S={s1,s2,...,sn} be a set of sites in Ed, where every site si has a positive real weight ωi. This paper gives algorithms to find weighted orthogonal L∞ and L1 approximating hyperplanes for S. The algorithm for the weighted orthogonal L1 approximation is shown to require O(nd) worst-case time and O(n) space for d ≥ 2. The algorithm for the weighted orthogonal L∞ approximation is shown to require O(n log n) worst-case time and O(n) space for d = 2, and O(n⌊dl2 + 1⌋) worst-case time and O(n⌊(d+1)/2⌋) space for d > 2. In the latter case, the expected time complexity may be reduced to O(n⌊(d+1)/2⌋). The L∞ approximation algorithm can be modified to solve the problem of finding the width of a set of n points in Ed, and the problem of finding a stabbing hyperplane for a set of n hyperspheres in Ed with varying radii. The time and space complexities of the width and stabbing algorithms are seen to be the same as those of the L∞ approximation algorithm
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
A weighted reduced basis method for parabolic PDEs with random data
This work considers a weighted POD-greedy method to estimate statistical
outputs parabolic PDE problems with parametrized random data. The key idea of
weighted reduced basis methods is to weight the parameter-dependent error
estimate according to a probability measure in the set-up of the reduced space.
The error of stochastic finite element solutions is usually measured in a root
mean square sense regarding their dependence on the stochastic input
parameters. An orthogonal projection of a snapshot set onto a corresponding POD
basis defines an optimum reduced approximation in terms of a Monte Carlo
discretization of the root mean square error. The errors of a weighted
POD-greedy Galerkin solution are compared against an orthogonal projection of
the underlying snapshots onto a POD basis for a numerical example involving
thermal conduction. In particular, it is assessed whether a weighted POD-greedy
solutions is able to come significantly closer to the optimum than a
non-weighted equivalent. Additionally, the performance of a weighted POD-greedy
Galerkin solution is considered with respect to the mean absolute error of an
adjoint-corrected functional of the reduced solution.Comment: 15 pages, 4 figure
Option Pricing with Orthogonal Polynomial Expansions
We derive analytic series representations for European option prices in
polynomial stochastic volatility models. This includes the Jacobi, Heston,
Stein-Stein, and Hull-White models, for which we provide numerical case
studies. We find that our polynomial option price series expansion performs as
efficiently and accurately as the Fourier transform based method in the nested
affine cases. We also derive and numerically validate series representations
for option Greeks. We depict an extension of our approach to exotic options
whose payoffs depend on a finite number of prices.Comment: forthcoming in Mathematical Finance, 38 pages, 3 tables, 7 figure
Sparse approximation of multivariate functions from small datasets via weighted orthogonal matching pursuit
We show the potential of greedy recovery strategies for the sparse
approximation of multivariate functions from a small dataset of pointwise
evaluations by considering an extension of the orthogonal matching pursuit to
the setting of weighted sparsity. The proposed recovery strategy is based on a
formal derivation of the greedy index selection rule. Numerical experiments
show that the proposed weighted orthogonal matching pursuit algorithm is able
to reach accuracy levels similar to those of weighted minimization
programs while considerably improving the computational efficiency for small
values of the sparsity level
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